NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
← Back to practice

Geometric construction algorithms

Wide tables scroll horizontally. For detailed working, you may prefer a larger screen.

Required skills (click to reveal)

In a community garden in Melbourne, a rectangular plot is designed with sides of length 21 m and 19 m. An algorithm is created to construct the diagonal of this rectangle using geometric principles, specifically by applying the Pythagorean theorem.

The algorithm includes a decision step to verify that the side lengths satisfy the triangle inequality for the right triangle formed by the diagonal.

What is the exact length of the diagonal, in metres, expressed in simplest radical form?

Concept illustrations

These are separate worked examples of related concepts, not a diagram of this question. Their values and dimensions do not describe your question.

Equal-radius circles locate a perpendicular bisector. Equal-radius circles centred at A and B meet above and below AB. The line through those intersections is the perpendicular bisector. The radius exceeds half the distance AB.ABEqual radii
Equal-radius circles centred at A and B meet above and below AB. The line through those intersections is the perpendicular bisector. The radius exceeds half the distance AB.
An even-number decision. Divide integer n by 2. If the remainder is 0, n is even; otherwise it is odd. The diagram is an algorithm, not a statement about non-integer inputs.Integer nRemainder of n ÷ 2= 0?YesNoEvenOdd
Divide integer n by 2. If the remainder is 0, n is even; otherwise it is odd. The diagram is an algorithm, not a statement about non-integer inputs.

Press Start when ready. Time includes time spent in another tab. Submit your answer before opening a help panel for a timed rank.

00:00
0 wordsOnly your numerical final answer or selected MCQ option is checked. Scratch working is not graded.

Report a problem or suggest an improvement

My notes — optional, not graded 0 words

Notes and scratch working stay in this browser and are linked to this question version. They are not graded or included in feedback. On shared devices, other people using this browser may see them.

For this question only. Your full notebook is on the dashboard.

Notes save in this browser. Download a copy to keep them elsewhere.

Report a problem or suggest an improvement

Your observations help improve this question. You can report an issue or recommend a better explanation, visual or experience.

0 words

The report includes your message and question/version details. Saved profile details, answers, notes and attempt history are not automatically attached. Leave out names and contact details, and check screenshots for personal information and embedded metadata.

Attachments may contain personal information. Review them before sharing. Logs collect only event codes and timestamps after you enable them. Download the ZIP to keep a copy, then attach it to an email if you choose.

Formula and explanation (click to reveal)

Opening this before your first submission makes the attempt practice only.

Formula

Explanation

Worked solution and answer (click to reveal)

Opening this before your first submission makes the attempt practice only.

Worked solution

  1. The rectangle has sides of 21 m and 19 m. The diagonal forms a right-angled triangle with these two sides.
  2. The Pythagorean theorem applies because the diagonal creates a right triangle with the two sides of the rectangle.
  3. The formula for the diagonal d is d² = a² + b², where a and b are the side lengths of the rectangle.
  4. Substitute a = 21 and b = 19 into the formula: d² = 21² + 19².
  5. Calculate 21²: 21 × 21 = 441.
  6. Calculate 19²: 19 × 19 = 361.
  7. Add the squares: 441 + 361 = 802.
  8. So d² = 802. Therefore, d = sqrt(802).
  9. Factor 802 to check for perfect square factors: 802 = 2 × 401. 401 is a prime number.
  10. Since 802 has no square factors other than 1, sqrt(802) cannot be simplified further.
  11. The diagonal length is exactly sqrt(802) metres.
  12. The triangle inequality is satisfied because 21 + 19 > sqrt(802), 21 + sqrt(802) > 19, and 19 + sqrt(802) > 21, as all sides are positive and the diagonal is less than the sum of the other two sides.

Reference answer: sqrt(802) m

Alternative methods (click to reveal)

Shortcuts are shown only when applicable. Check the conditions before using one.

Earlier attempts for this question

#Date / timeYour answerResultSecondsHelp revealedRank
History stays in this browser when storage is available. Export it to your shared Drive folder and import it on another computer.
Worksheet details

Naming pattern: Topic_Grade_Complexity_YYYYMMDD_HH24MISS.html

Validation compares your answer with the generated reference. The model's reference can be wrong. Time ranks use an estimated student solving time, not a measured benchmark or a comparison with other students. An incorrect answer receives no speed rank.

NavGanitMate privacy & saved work

Saved in this browser. Your profile preferences, answers, practice history and notes are stored here when browser storage is available. They are not automatically uploaded to our servers. People sharing this browser may see saved work; changing the learner name does not create a separate private account. A nickname is enough.

Keep copies of important work. Clearing browser data, private browsing or changing browser or device can mean losing saved work. Download notes and export history separately. Downloads are copies of the selected content, not a complete account backup; not every format can be imported to restore your work. Keep these files somewhere private.

Visiting the website. Cloudflare helps deliver and protect this site and processes connection information such as IP addresses and requested pages. Server error logs may also contain request details. Security features may use cookies. This is separate from your browser-saved learning records. Cloudflare privacy information.

A parent-connection reminder uses a local list of up to five completed question IDs for this browser profile and a tab-session dismissal marker. Reminder activity is not uploaded and practice remains available.

Sharing feedback is your choice. A report includes your message and question/version details. Your saved profile, answers and notes are not automatically attached. Files you select can reveal personal details or contain embedded metadata: check and redact them before sharing. Diagnostic event codes and timestamps are included only after you enable logging. Sending email shares your message and attachments with the recipient and email providers. A download alone sends nothing to us. Website feedback delivery may be unavailable; only a confirmed receipt indicates it was received.